two isomorphic groups

The set of 3⁢x⁡3permutation matrices form a group under matrix multiplication. This example demonstrates that fact and develops the multiplication table and compares it to S3. Although there are alternative ways to fill in the table, this example serves to help the beginner. Here we will see that the two groups have the
same structure. We begin by defining the elements of our group.

Here, our group is just P3={I,A,B,R,S,T}. Now, we can start to multiply and then fill in the table.
First, we calculate the square of each elements.

A2=(010001100)⁢(010001100)=(001100010)=B

B2=(001100010)⁢(001100010)=(010001100)=A

R2=(100001010)⁢(100001010)=(100010001)=I

S2=(001010100)⁢(001010100)=(100010001)=I

T2=(010100001)⁢(010100001)=(100010001)=I

Now starting with the upper left 3⁢x⁡3 block, we go through the table.

A⁢B=(010001100)⁢(001100010)=(100010001)=I

B⁢A=(001100010)⁢(010001100)=(100010001)=I

We can complete the upper left 3⁢x⁡3 block of the table and complete diagonal using the above values. We note that no row or column can have a repeated elements which follows from the of a group. Next, we work on the upper right 3⁢x⁡3 block of the table.

A⁢R=(010001100)⁢(100001010)=(001010100)=S

A⁢S=(010001100)⁢(001010100)=(010100001)=T

Now we can complete the upper right 3 x 3 block of the table. Next, we work on the lower left 3⁢x⁡3 block of the table.

R⁢A=(100001010)⁢(010001100)=(010100001)=T

R⁢B=(100001010)⁢(001100010)=(001010100)=S

S⁢A=(001010100)⁢(010001100)=(100001010)=R

Now we can complete the lower left 3⁢x⁡3 block of the table. Finally, we work on the lower right 3⁢x⁡3 block of the table.